Local model predictive control experiences with differential driven wheeled mobile robots

This work extends a previously developed research concerning about the use of local model predictive control in differential driven mobile robots. Hence, experimental results are presented as a way to improve the methodology by considering aspects as trajectory accuracy and time performance. In this sense, the cost function and the prediction horizon are important aspects to be considered. The aim of the present work is to test the control method by measuring trajectory tracking accuracy and time performance. Moreover, strategies for the integration with perception system and path planning are briefly introduced. In this sense, monocular image data can be used to plan safety trajectories by using goal attraction potential fields

Mobile Robot Local Predictive Control under Perception Constraints

This research extends a previously developed work concerning about the use of local model predictive control in mobile robots. Hence, experimental results are presented as a way to improve the methodology by considering aspects as trajectory accuracy and time performance. In this sense, the cost function and the prediction horizon are important aspects to be considered. The platformused is a differential driven robot with a free rotating wheel. The aim of the present work is to test the control method by measuring trajectory tracking accuracy and time performance. Moreover, strategies for the integration with perception system and path planning are also introduced. In this sense, monocular image data provide an occupancy grid where safety trajectories are computed by using goal attraction potential fields

USE OF A LOW-COST TEMPLATE-FREE ZSM-5 FOR ATMOSPHERIC PETROLEUM RESIDUE PYROLYSIS

To understand the physicochemical properties and catalytic activity during the pyrolysis of atmospheric petroleum residue, a template-free ZSM-5 zeolite was synthesized using a direct method without additional seeds or an organic structure director and compared with conventionally synthesized ZSM-5. The crystallinities of the two zeolites were evaluated by XRD and FTIR and were quite similar; however, structural analyses using SEM and argon physisorption revealed that the zeolites diverged in particle diameter and in the external surface area of the micropores. The synthesis procedure without a template incorporated additional aluminum into the crystalline network, according to ICP-AES and TPD NH3 experiments. The catalytic pyrolysis performed over the template-free ZSM-5 generated results comparable to those for pyrolysis performed over the conventional ZSM-5 according to its hydrocarbon distribution. The selectivity to aromatics compounds was exactly the same for both ZSM-5 zeolites, and these values stand out compared to thermal pyrolysis. The template-free ZSM-5 produced 20% of light hydrocarbons (C4-C6), where such compounds are olefins and paraffins of great interest to the petrochemical industry. Therefore, template-free ZSM-5 is promising for industrial use due to its lowered synthesis time, low-cost and significant distribution to light hydrocarbons.

MCMC Analysis for Optimization of Stochastic Models

In any decision making under uncertainties, the goal is mostly to minimize the expected cost. The minimization of cost under uncertainties is usually done by optimization. For simple models, the optimization can easily be done using deterministic methods.However, many models practically contain some complex and varying parameters that can not easily be taken into account using usual deterministic methods of optimization. Thus, it is very important to look for other methods that can be used to get insight into such models. MCMC method is one of the practical methods that can be used for optimization of stochastic models under uncertainty. This method is based on simulation that provides a general methodology which can be applied in nonlinear and non-Gaussian state models. MCMC method is very important for practical applications because it is a uni ed estimation procedure which simultaneously estimates both parameters and state variables. MCMC computes the distribution of the state variables and parameters of the given data measurements. MCMC method is faster in terms of computing time when compared to other optimization methods. This thesis discusses the use of Markov chain Monte Carlo (MCMC) methods for optimization of Stochastic models under uncertainties .The thesis begins with a short discussion about Bayesian Inference, MCMC and Stochastic optimization methods. Then an example is given of how MCMC can be applied for maximizing production at a minimum cost in a chemical reaction process. It is observed that this method performs better in optimizing the given cost function with a very high certainty.

Voimalaitoksen käytönaikaisen optimoinnin määrittely

Tässä diplomityössä määritellään biopolttoainetta käyttävän voimalaitoksen käytönaikainen tuotannon optimointimenetelmä. Määrittelytyö liittyy MW Powerin MultiPower CHP –voimalaitoskonseptin jatkokehitysprojektiin. Erilaisten olemassa olevien optimointitapojen joukosta valitaan tarkoitukseen sopiva, laitosmalliin ja kustannusfunktioon perustuva menetelmä, jonka tulokset viedään automaatiojärjestelmään PID-säätimien asetusarvojen muodossa. Prosessin mittaustulosten avulla lasketaan laitoksen energia- ja massataseet, joiden tuloksia käytetään seuraavan optimointihetken lähtötietoina. Optimoinnin kohdefunktio on kustannusfunktio, jonka termit ovat voimalaitoksen käytöstä aiheutuvia tuottoja ja kustannuksia. Prosessia optimoidaan säätimille annetut raja-arvot huomioiden niin, että kokonaiskate maksimoituu. Kun laitokselle kertyy käyttöikää ja historiadataa, voidaan prosessin optimointia nopeuttaa hakemalla tilastollisesti historiadatasta nykytilanteen olosuhteita vastaava hetki. Kyseisen historian hetken katetta verrataan kustannusfunktion optimoinnista saatuun katteeseen. Paremman katteen antavan menetelmän laskemat asetusarvot otetaan käyttöön prosessin ohjausta varten. Mikäli kustannusfunktion laskenta eikä historiadatan perusteella tehty haku anna paranevaa katetta, niiden laskemia asetusarvoja ei oteta käyttöön. Sen sijaan optimia aletaan hakea deterministisellä optimointialgoritmilla, joka hakee nykyhetken ympäristöstä paremman katteen antavia säätimien asetusarvoja. Säätöjärjestelmä on mahdollista toteuttaa myös tulevaisuutta ennustavana. Työn käytännön osuudessa voimalaitosmalli luodaan kahden eri mallinnusohjelman avulla, joista toisella kuvataan kattilan ja toisella voimalaitosprosessin toimintaa. Mallinnuksen tuloksena saatuja prosessiarvoja hyödynnetään lähtötietoina käyttökatteen laskennassa. Kate lasketaan kustannusfunktion perusteella. Tuotoista suurimmat liittyvät sähkön ja lämmön myyntiin sekä tuotantotukeen, ja suurimmat kustannukset liittyvät investoinnin takaisinmaksuun ja polttoaineen ostoon. Kustannusfunktiolle tehdään herkkyystarkastelu, jossa seurataan katteen muutosta prosessin teknisiä arvoja muutettaessa. Tuloksia vertaillaan referenssivoimalaitoksella suoritettujen verifiointimittausten tuloksiin, ja havaitaan, että tulokset eivät ole täysin yhteneviä. Erot johtuvat sekä mallinnuksen puutteista että mittausten lyhyehköistä tarkasteluajoista. Automatisoidun optimointijärjestelmän käytännön toteutusta alustetaan määrittelemällä käyttöön otettava optimointitapa, siihen liittyvät säätöpiirit ja tarvittavat lähtötiedot. Projektia tullaan jatkamaan järjestelmän ohjelmoinnilla, testauksella ja virityksellä todellisessa voimalaitosympäristössä ja myöhemmin ennustavan säädön toteuttamisella.

Estimation of aircraft aerodynamic derivatives using Extended Kalman Filter

Design of flight control laws, verification of performance predictions, and the implementation of flight simulations are tasks that require a mathematical model of the aircraft dynamics. The dynamical models are characterized by coefficients (aerodynamic derivatives) whose values must be determined from flight tests. This work outlines the use of the Extended Kalman Filter (EKF) in obtaining the aerodynamic derivatives of an aircraft. The EKF shows several advantages over the more traditional least-square method (LS). Among these the most important are: there are no restrictions on linearity or in the form which the parameters appears in the mathematical model describing the system, and it is not required that these parameters be time invariant. The EKF uses the statistical properties of the process and the observation noise, to produce estimates based on the mean square error of the estimates themselves. Differently, the LS minimizes a cost function based on the plant output behavior. Results for the estimation of some longitudinal aerodynamic derivatives from simulated data are presented.

Responsibility and Cross-Subsidization in Cost Sharing

We propose two axiomatic theories of cost sharing with the common premise that agents demand comparable -though perhaps different- commodities and are responsible for their own demand. Under partial responsibility the agents are not responsible for the asymmetries of the cost function: two agents consuming the same amount of output always pay the same price; this holds true under full responsibility only if the cost function is symmetric in all individual demands. If the cost function is additively separable, each agent pays her stand alone cost under full responsibility; this holds true under partial responsibility only if, in addition, the cost function is symmetric. By generalizing Moulin and Shenker’s (1999) Distributivity axiom to cost-sharing methods for heterogeneous goods, we identify in each of our two theories a different serial method. The subsidy-free serial method (Moulin, 1995) is essentially the only distributive method meeting Ranking and Dummy. The cross-subsidizing serial method (Sprumont, 1998) is the only distributive method satisfying Separability and Strong Ranking. Finally, we propose an alternative characterization of the latter method based on a strengthening of Distributivity.

Étude de l’implication de MK2 dans la réponse vasculaire à l’endothéline-1

L’endothéline-1 (ET-1) est un puissant agent vasoconstricteur dont la production est dérégulée dans plusieurs maladies inflammatoires où l’expression des cyclooxygénases-1/2 (COX-1/2) est augmentée. Puisqu’il est connu que la voie p38 MAPK est impliquée dans la régulation de l’ET-1 au niveau de l’ARNm, nous avons étudié le rôle de l’un de ses substrats, la kinase MK2 dans la régulation post-transcriptionnelle de l’ET-1 et des COX. Pour ce faire, nous avons utilisé des souris MK2-déficientes (MK2-/-) ainsi que des contrôles (MK2+/+) issus de la même portée. Des paramètres de la fonction cardiaque ont été mesurés sous anesthésie à l’aide d’un cathéter Millar et la réactivité vasculaire de l’artère fémorale a été mesurée par myographe. L’expression de ET-1, COX-1 et COX-2 a été quantifiée dans la cellule endothéliale aortique (CE) par qPCR. En réponse à l’ET-1 (100 nM), l’expression de la préproET-1 dans les CE augmente en fonction du temps (p<0.05) : cette variation est accentuée chez les souris MK2-/-. Bien que la pression artérielle soit similaire entre les souris MK2+/+ et MK2-/-, l’inhibition de COX (indométacine, 1 μM) augmente (p<0.05) la contraction à l’ET-1 des vaisseaux isolés provenant de souris MK2+/+ mais pas des MK2-/-. Ces données suggèrent un rôle de MK2 dans la réponse vasculaire à l’ET-1 et possiblement dans la signalisation post-récepteur de l’ET-1 en général.

Problèmes de premier passage et de commande optimale pour des chaînes de Markov à temps discret.

Nous considérons des processus de diffusion, définis par des équations différentielles stochastiques, et puis nous nous intéressons à des problèmes de premier passage pour les chaînes de Markov en temps discret correspon- dant à ces processus de diffusion. Comme il est connu dans la littérature, ces chaînes convergent en loi vers la solution des équations différentielles stochas- tiques considérées. Notre contribution consiste à trouver des formules expli- cites pour la probabilité de premier passage et la durée de la partie pour ces chaînes de Markov à temps discret. Nous montrons aussi que les résultats ob- tenus convergent selon la métrique euclidienne (i.e topologie euclidienne) vers les quantités correspondantes pour les processus de diffusion. En dernier lieu, nous étudions un problème de commande optimale pour des chaînes de Markov en temps discret. L’objectif est de trouver la valeur qui mi- nimise l’espérance mathématique d’une certaine fonction de coût. Contraire- ment au cas continu, il n’existe pas de formule explicite pour cette valeur op- timale dans le cas discret. Ainsi, nous avons étudié dans cette thèse quelques cas particuliers pour lesquels nous avons trouvé cette valeur optimale.

Problèmes de commande optimale stochastique généralisés

Cette thèse est divisée en deux grands chapitres, dont le premier porte sur des problèmes de commande optimale en dimension un et le deuxième sur des problèmes en dimension deux ou plus. Notons bien que, dans cette thèse, nous avons supposé que le facteur temps n'intervient pas. Dans le premier chapitre, nous calculons, au début, l'équation de programmation dynamique pour la valeur minimale F de l'espérance mathématique de la fonction de coût considérée. Ensuite, nous utilisons le théorème de Whittle qui est applicable seulement si une condition entre le bruit blanc v et les termes b et q associés à la commande est satisfaite. Sinon, nous procédons autrement. En effet, un changement de variable transforme notre équation en une équation de Riccati en G= F', mais sans conditions initiales. Dans certains cas, à partir de la symétrie des paramètres infinitésimaux et de q, nous pouvons en déduire le point x' où G(x')=0. Si ce n'est pas le cas, nous nous limitons à des bonnes approximations. Cette même démarche est toujours possible si nous sommes dans des situations particulières, par exemple, lorsque nous avons une seule barrière. Dans le deuxième chapitre, nous traitons les problèmes en dimension deux ou plus. Puisque la condition de Whittle est difficile à satisfaire dans ce cas, nous essayons de généraliser les résultats du premier chapitre. Nous utilisons alors dans quelques exemples la méthode des similitudes, qui permet de transformer le problème en dimension un. Ensuite, nous proposons une nouvelle méthode de résolution. Cette dernière linéarise l'équation de programmation dynamique qui est une équation aux dérivées partielles non linéaire. Il reste à la fin à trouver les conditions initiales pour la nouvelle fonction et aussi à vérifier que les n expressions obtenues pour F sont équivalentes.

Optimierung von Betriebsstrategien für elektrifizierte Antriebskonzepte

Im Rahmen dieser Arbeit wird eine gemeinsame Optimierung der Hybrid-Betriebsstrategie und des Verhaltens des Verbrennungsmotors vorgestellt. Die Übernahme von den im Steuergerät verwendeten Funktionsmodulen in die Simulationsumgebung für Fahrzeuglängsdynamik stellt eine effiziente Applikationsmöglichkeit der Originalparametrierung dar. Gleichzeitig ist es notwendig, das Verhalten des Verbrennungsmotors derart nachzubilden, dass das stationäre und das dynamische Verhalten, inklusive aller relevanten Einflussmöglichkeiten, wiedergegeben werden kann. Das entwickelte Werkzeug zur Übertragung der in Ascet definierten Steurgerätefunktionen in die Simulink-Simulationsumgebung ermöglicht nicht nur die Simulation der relevanten Funktionsmodule, sondern es erfüllt auch weitere wichtige Eigenschaften. Eine erhöhte Flexibilität bezüglich der Daten- und Funktionsstandänderungen, sowie die Parametrierbarkeit der Funktionsmodule sind Verbesserungen die an dieser Stelle zu nennen sind. Bei der Modellierung des stationären Systemverhaltens des Verbrennungsmotors erfolgt der Einsatz von künstlichen neuronalen Netzen. Die Auswahl der optimalen Neuronenanzahl erfolgt durch die Betrachtung des SSE für die Trainings- und die Verifikationsdaten. Falls notwendig, wird zur Sicherstellung der angestrebten Modellqualität, das Interpolationsverhalten durch Hinzunahme von Gauß-Prozess-Modellen verbessert. Mit den Gauß-Prozess-Modellen werden hierbei zusätzliche Stützpunkte erzeugt und mit einer verminderten Priorität in die Modellierung eingebunden. Für die Modellierung des dynamischen Systemverhaltens werden lineare Übertragungsfunktionen verwendet. Bei der Minimierung der Abweichung zwischen dem Modellausgang und den Messergebnissen wird zusätzlich zum SSE das 2σ-Intervall der relativen Fehlerverteilung betrachtet. Die Implementierung der Steuergerätefunktionsmodule und der erstellten Steller-Sensor-Streckenmodelle in der Simulationsumgebung für Fahrzeuglängsdynamik führt zum Anstieg der Simulationszeit und einer Vergrößerung des Parameterraums. Das aus Regelungstechnik bekannte Verfahren der Gütevektoroptimierung trägt entscheidend zu einer systematischen Betrachtung und Optimierung der Zielgrößen bei. Das Ergebnis des Verfahrens ist durch das Optimum der Paretofront der einzelnen Entwurfsspezifikationen gekennzeichnet. Die steigenden Simulationszeiten benachteiligen Minimumsuchverfahren, die eine Vielzahl an Iterationen benötigen. Um die Verwendung einer Zufallsvariablen, die maßgeblich zur Steigerung der Iterationanzahl beiträgt, zu vermeiden und gleichzeitig eine Globalisierung der Suche im Parameterraum zu ermöglichen wird die entwickelte Methode DelaunaySearch eingesetzt. Im Gegensatz zu den bekannten Algorithmen, wie die Partikelschwarmoptimierung oder die evolutionären Algorithmen, setzt die neu entwickelte Methode bei der Suche nach dem Minimum einer Kostenfunktion auf eine systematische Analyse der durchgeführten Simulationsergebnisse. Mit Hilfe der bei der Analyse gewonnenen Informationen werden Bereiche mit den bestmöglichen Voraussetzungen für ein Minimum identifiziert. Somit verzichtet das iterative Verfahren bei der Bestimmung des nächsten Iterationsschrittes auf die Verwendung einer Zufallsvariable. Als Ergebnis der Berechnungen steht ein gut gewählter Startwert für eine lokale Optimierung zur Verfügung. Aufbauend auf der Simulation der Fahrzeuglängsdynamik, der Steuergerätefunktionen und der Steller-Sensor-Streckenmodelle in einer Simulationsumgebung wird die Hybrid-Betriebsstrategie gemeinsam mit der Steuerung des Verbrennungsmotors optimiert. Mit der Entwicklung und Implementierung einer neuen Funktion wird weiterhin die Verbindung zwischen der Betriebsstrategie und der Motorsteuerung erweitert. Die vorgestellten Werkzeuge ermöglichten hierbei nicht nur einen Test der neuen Funktionalitäten, sondern auch eine Abschätzung der Verbesserungspotentiale beim Verbrauch und Abgasemissionen. Insgesamt konnte eine effiziente Testumgebung für eine gemeinsame Optimierung der Betriebsstrategie und des Verbrennungsmotorverhaltens eines Hybridfahrzeugs realisiert werden.

Local model predictive control experiences with differential driven wheeled mobile robots

This work extends a previously developed research concerning about the use of local model predictive control in differential driven mobile robots. Hence, experimental results are presented as a way to improve the methodology by considering aspects as trajectory accuracy and time performance. In this sense, the cost function and the prediction horizon are important aspects to be considered. The aim of the present work is to test the control method by measuring trajectory tracking accuracy and time performance. Moreover, strategies for the integration with perception system and path planning are briefly introduced. In this sense, monocular image data can be used to plan safety trajectories by using goal attraction potential fields

Modelling, control and supervision for a class of hybrid systems

The aim of this thesis is to narrow the gap between two different control techniques: the continuous control and the discrete event control techniques DES. This gap can be reduced by the study of Hybrid systems, and by interpreting as Hybrid systems the majority of large-scale systems. In particular, when looking deeply into a process, it is often possible to identify interaction between discrete and continuous signals. Hybrid systems are systems that have both continuous, and discrete signals. Continuous signals are generally supposed continuous and differentiable in time, since discrete signals are neither continuous nor differentiable in time due to their abrupt changes in time. Continuous signals often represent the measure of natural physical magnitudes such as temperature, pressure etc. The discrete signals are normally artificial signals, operated by human artefacts as current, voltage, light etc. Typical processes modelled as Hybrid systems are production systems, chemical process, or continuos production when time and continuous measures interacts with the transport, and stock inventory system. Complex systems as manufacturing lines are hybrid in a global sense. They can be decomposed into several subsystems, and their links. Another motivation for the study of Hybrid systems is the tools developed by other research domains. These tools benefit from the use of temporal logic for the analysis of several properties of Hybrid systems model, and use it to design systems and controllers, which satisfies physical or imposed restrictions. This thesis is focused in particular types of systems with discrete and continuous signals in interaction. That can be modelled hard non-linealities, such as hysteresis, jumps in the state, limit cycles, etc. and their possible non-deterministic future behaviour expressed by an interpretable model description. The Hybrid systems treated in this work are systems with several discrete states, always less than thirty states (it can arrive to NP hard problem), and continuous dynamics evolving with expression: with Ki ¡ Rn constant vectors or matrices for X components vector. In several states the continuous evolution can be several of them Ki = 0. In this formulation, the mathematics can express Time invariant linear system. By the use of this expression for a local part, the combination of several local linear models is possible to represent non-linear systems. And with the interaction with discrete events of the system the model can compose non-linear Hybrid systems. Especially multistage processes with high continuous dynamics are well represented by the proposed methodology. Sate vectors with more than two components, as third order models or higher is well approximated by the proposed approximation. Flexible belt transmission, chemical reactions with initial start-up and mobile robots with important friction are several physical systems, which profits from the benefits of proposed methodology (accuracy). The motivation of this thesis is to obtain a solution that can control and drive the Hybrid systems from the origin or starting point to the goal. How to obtain this solution, and which is the best solution in terms of one cost function subject to the physical restrictions and control actions is analysed. Hybrid systems that have several possible states, different ways to drive the system to the goal and different continuous control signals are problems that motivate this research. The requirements of the system on which we work is: a model that can represent the behaviour of the non-linear systems, and that possibilities the prediction of possible future behaviour for the model, in order to apply an supervisor which decides the optimal and secure action to drive the system toward the goal. Specific problems can be determined by the use of this kind of hybrid models are: - The unity of order. - Control the system along a reachable path. - Control the system in a safe path. - Optimise the cost function. - Modularity of control The proposed model solves the specified problems in the switching models problem, the initial condition calculus and the unity of the order models. Continuous and discrete phenomena are represented in Linear hybrid models, defined with defined eighth-tuple parameters to model different types of hybrid phenomena. Applying a transformation over the state vector : for LTI system we obtain from a two-dimensional SS a single parameter, alpha, which still maintains the dynamical information. Combining this parameter with the system output, a complete description of the system is obtained in a form of a graph in polar representation. Using Tagaki-Sugeno type III is a fuzzy model which include linear time invariant LTI models for each local model, the fuzzyfication of different LTI local model gives as a result a non-linear time invariant model. In our case the output and the alpha measure govern the membership function. Hybrid systems control is a huge task, the processes need to be guided from the Starting point to the desired End point, passing a through of different specific states and points in the trajectory. The system can be structured in different levels of abstraction and the control in three layers for the Hybrid systems from planning the process to produce the actions, these are the planning, the process and control layer. In this case the algorithms will be applied to robotics ¡V a domain where improvements are well accepted ¡V it is expected to find a simple repetitive processes for which the extra effort in complexity can be compensated by some cost reductions. It may be also interesting to implement some control optimisation to processes such as fuel injection, DC-DC converters etc. In order to apply the RW theory of discrete event systems on a Hybrid system, we must abstract the continuous signals and to project the events generated for these signals, to obtain new sets of observable and controllable events. Ramadge & Wonham¡¦s theory along with the TCT software give a Controllable Sublanguage of the legal language generated for a Discrete Event System (DES). Continuous abstraction transforms predicates over continuous variables into controllable or uncontrollable events, and modifies the set of uncontrollable, controllable observable and unobservable events. Continuous signals produce into the system virtual events, when this crosses the bound limits. If this event is deterministic, they can be projected. It is necessary to determine the controllability of this event, in order to assign this to the corresponding set, , controllable, uncontrollable, observable and unobservable set of events. Find optimal trajectories in order to minimise some cost function is the goal of the modelling procedure. Mathematical model for the system allows the user to apply mathematical techniques over this expression. These possibilities are, to minimise a specific cost function, to obtain optimal controllers and to approximate a specific trajectory. The combination of the Dynamic Programming with Bellman Principle of optimality, give us the procedure to solve the minimum time trajectory for Hybrid systems. The problem is greater when there exists interaction between adjacent states. In Hybrid systems the problem is to determine the partial set points to be applied at the local models. Optimal controller can be implemented in each local model in order to assure the minimisation of the local costs. The solution of this problem needs to give us the trajectory to follow the system. Trajectory marked by a set of set points to force the system to passing over them. Several ways are possible to drive the system from the Starting point Xi to the End point Xf. Different ways are interesting in: dynamic sense, minimum states, approximation at set points, etc. These ways need to be safe and viable and RchW. And only one of them must to be applied, normally the best, which minimises the proposed cost function. A Reachable Way, this means the controllable way and safe, will be evaluated in order to obtain which one minimises the cost function. Contribution of this work is a complete framework to work with the majority Hybrid systems, the procedures to model, control and supervise are defined and explained and its use is demonstrated. Also explained is the procedure to model the systems to be analysed for automatic verification. Great improvements were obtained by using this methodology in comparison to using other piecewise linear approximations. It is demonstrated in particular cases this methodology can provide best approximation. The most important contribution of this work, is the Alpha approximation for non-linear systems with high dynamics While this kind of process is not typical, but in this case the Alpha approximation is the best linear approximation to use, and give a compact representation.

The full-spectrum correlated-k method for longwave atmospheric radiative transfer using an effective Planck function

The correlated k-distribution (CKD) method is widely used in the radiative transfer schemes of atmospheric models and involves dividing the spectrum into a number of bands and then reordering the gaseous absorption coefficients within each one. The fluxes and heating rates for each band may then be computed by discretizing the reordered spectrum into of order 10 quadrature points per major gas and performing a monochromatic radiation calculation for each point. In this presentation it is shown that for clear-sky longwave calculations, sufficient accuracy for most applications can be achieved without the need for bands: reordering may be performed on the entire longwave spectrum. The resulting full-spectrum correlated k (FSCK) method requires significantly fewer monochromatic calculations than standard CKD to achieve a given accuracy. The concept is first demonstrated by comparing with line-by-line calculations for an atmosphere containing only water vapor, in which it is shown that the accuracy of heating-rate calculations improves approximately in proportion to the square of the number of quadrature points. For more than around 20 points, the root-mean-squared error flattens out at around 0.015 K/day due to the imperfect rank correlation of absorption spectra at different pressures in the profile. The spectral overlap of m different gases is treated by considering an m-dimensional hypercube where each axis corresponds to the reordered spectrum of one of the gases. This hypercube is then divided up into a number of volumes, each approximated by a single quadrature point, such that the total number of quadrature points is slightly fewer than the sum of the number that would be required to treat each of the gases separately. The gaseous absorptions for each quadrature point are optimized such that they minimize a cost function expressing the deviation of the heating rates and fluxes calculated by the FSCK method from line-by-line calculations for a number of training profiles. This approach is validated for atmospheres containing water vapor, carbon dioxide, and ozone, in which it is found that in the troposphere and most of the stratosphere, heating-rate errors of less than 0.2 K/day can be achieved using a total of 23 quadrature points, decreasing to less than 0.1 K/day for 32 quadrature points. It would be relatively straightforward to extend the method to include other gases.

Estimation of optimal gravity wave parameters for climate models using data assimilation

There is a current need to constrain the parameters of gravity wave drag (GWD) schemes in climate models using observational information instead of tuning them subjectively. In this work, an inverse technique is developed using data assimilation principles to estimate gravity wave parameters. Because mostGWDschemes assume instantaneous vertical propagation of gravity waves within a column, observations in a single column can be used to formulate a one-dimensional assimilation problem to estimate the unknown parameters. We define a cost function that measures the differences between the unresolved drag inferred from observations (referred to here as the ‘observed’ GWD) and the GWD calculated with a parametrisation scheme. The geometry of the cost function presents some difficulties, including multiple minima and ill-conditioning because of the non-independence of the gravity wave parameters. To overcome these difficulties we propose a genetic algorithm to minimize the cost function, which provides a robust parameter estimation over a broad range of prescribed ‘true’ parameters. When real experiments using an independent estimate of the ‘observed’ GWD are performed, physically unrealistic values of the parameters can result due to the non-independence of the parameters. However, by constraining one of the parameters to lie within a physically realistic range, this degeneracy is broken and the other parameters are also found to lie within physically realistic ranges. This argues for the essential physical self-consistency of the gravity wave scheme. A much better fit to the observed GWD at high latitudes is obtained when the parameters are allowed to vary with latitude. However, a close fit can be obtained either in the upper or the lower part of the profiles, but not in both at the same time. This result is a consequence of assuming an isotropic launch spectrum. The changes of sign in theGWDfound in the tropical lower stratosphere, which are associated with part of the quasi-biennial oscillation forcing, cannot be captured by the parametrisation with optimal parameters.